`"If "y=(2)/(sqrt(a^(2)-b^(2))){tan^(-1)(sqrt((a-b)/(a+b))tan""(x)/(2))}," then show that "(d^(2)y)/(dx^(2))=(b sin x)/((a+b cos x)^(2)).`

Explanation

`y=A tan^(-1)(B tan""(x)/(2)),`
`"where "A=(2)/(sqrt(a^(2)-b^(2))),B=sqrt((a-b)/(a+b))`
`Ab=(2)/(sqrt((a-b)(a+b)))sqrt((a-b)/(a+b))=(2)/(a+b)`
`(dy)/(dx)=(ABsec^(2)""(x)/(2)xx(1)/(2))/(1+B^(2)tan^(2)""(x)/(2))`
`=(1)/(a+b).(a+b)/((a+b)cos^(2)""(x)/(2)+(a-b)sin^(2)""(x)/(2))`
`=(1)/(a+b cos x)" (1)"`
`therefore" " (d^(2)y)/(dx^(2))=(b sin x)/((a+b cos x)^(2))`